zbMATH — the first resource for mathematics

On the real exponential field with restricted analytic functions. (English) Zbl 0823.03017
The present paper generalizes a result of A. Wilkie who has proved the model-completeness of the structure \(\mathbb{R}_{\exp}\), i.e. the field of reals together with exponentiation. In an earlier paper, Denef and van den Dries have already proved the model-completeness of \(\mathbb{R}_{\text{an}}\), the field of reals endowed for every \(m\geq 1\) with the restriction to \([- 1,+ 1]^ m\) of all \(m\)-ary real-valued functions \(f\), analytic in a neighbourhood of \([- 1, +1]^ m\). Now, this paper gives a proof of the model-completeness of the structure \((\mathbb{R}_{\text{an}},\exp)\), i.e. \(\mathbb{R}_{\text{an}}\) together with real exponentiation. The proof generalizes that of A. Wilkie and, in addition, gives o-minimality for the structure \((\mathbb{R}_{\text{an}}, \exp)\). There is a more recent work of the first author, A.Macintyre and D. Marker [Ann. Math., II. Ser. 140, 183-205 (1994)] which proves an even stronger result for \((\mathbb{R}_{\text{an}}, \exp)\) by using a different method in the proof.

03C60 Model-theoretic algebra
12L12 Model theory of fields
Full Text: DOI
[1] [D-vdD] J. Denef and L. van den Dries,P-adic and real subanalytic sets, Ann. Math.128 (1988), 79–138. · Zbl 0693.14012 · doi:10.2307/1971463
[2] [vdD1] L. van den Dries,Algebraic theories with definable Skolem functions, Journal of Symbolic Logic49 (1984), 625–629. · Zbl 0596.03032 · doi:10.2307/2274194
[3] [vdD2] L. van den Dries,A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. AMS15 (1986), 189–193. · Zbl 0612.03008 · doi:10.1090/S0273-0979-1986-15468-6
[4] [vdD3] L. van den Dries,The elementary theory of restricted elementary functions, J. Symb. Logic53 (1988), 796–808. · Zbl 0698.03023 · doi:10.2307/2274572
[5] [vdD-M-M] L. van den Dries, A. Macintyre and D. Marker,The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, to appear.
[6] [F] J. Frisch,Points de platitude d’un morphisme d’espaces analytiques, Inv. Math.4 (1967), 118–138. · Zbl 0167.06803 · doi:10.1007/BF01425245
[7] [H] A. G. Hovanskii,On a class of systems of transcendental equations, Soviet Math. Dokl.22 (1980), 762–765. · Zbl 0569.32004
[8] [K-P-S] J. Knight, A. Pillay and C. Steinhorn,Definable sets in ordered structures. II, Trans. AMS295 (1986), 593–605. · Zbl 0662.03024 · doi:10.1090/S0002-9947-1986-0833698-1
[9] [P-S] A. Pillay and C. Steinhorn,Definable sets in ordered structures. I, Trans. AMS295 (1986), 565–592. · Zbl 0662.03023 · doi:10.1090/S0002-9947-1986-0833697-X
[10] [Re] J.-P. Ressayre,Integer parts of real closed exponential fields, preprint. · Zbl 0791.03018
[11] [Ro] M. Rosenlicht,The rank of a Hardy field, Trans. AMS280 (1983), 659–671. · Zbl 0536.12015 · doi:10.1090/S0002-9947-1983-0716843-5
[12] [W1] A. J. Wilkie,Model completeness results for expansions of the real ordered field I: Restricted Pfaffian functions, preprint 1991.
[13] [W2] A. J. Wilkie,Model completeness results for expansions of the real field II: the exponential function, preprint 1992.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.